arXiv:2008.00944 · IOP SciNotes

Dynamics of Rényi entanglement entropy in diffusive qudit systems

Yichen Huang

Published Dec 1, 2020• 15 citations

Abstract

Abstract My previous work [arXiv: 1902.00977 ] studied the dynamics of Rényi entanglement entropy R α in local quantum circuits with charge conservation. Initializing the system in a random product state, it was proved that R α with Rényi index α > 1 grows no faster than ‘diffusively’ (up to a sublogarithmic correction) if charge transport is not faster than diffusive. The proof was given only for qubit or spin-1/2 systems. In this note, I extend the proof to qudit systems, i.e., spin systems with local dimension d ≥ 2 .

DOI: 10.1088/2633-1357/abd1e2

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arXiv:2008.00944 · IOP SciNotes

Dynamics of Rényi entanglement entropy in diffusive qudit systems

Yichen Huang

Published Dec 1, 2020• 15 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/2633-1357/abd1e2

Related papers

Dynamics of Renyi entanglement entropy in local quantum circuits with charge conservation

Yichen Huang

In local quantum circuits with charge conservation, we initialize the system in random product states and study the dynamics of the Renyi entanglement entropy $R_α$ . We rigorously prove that $R_α$ with Renyi index $α>1$ at time $t$ is $\le O(\sqrt{t\ln t})$ if the transport of charges is diffusive. Very recent numerical results of Rakovszky et al. show that this upper bound is saturated (up to the sub-logarithmic correction) in random local quantum circuits with charge conservation.In local quantum circuits with charge conservation, we initialize the system in random product states and study the dynamics of the Renyi entanglement entropy $R_α$. We rigorously prove that $R_α$ with Renyi index $α>1$ at time $t$ is $\le O(\sqrt{t\ln t})$ if the transport of charges is diffusive. Very recent numerical results of Rakovszky et al. show that this upper bound is saturated (up to the sub-logarithmic correction) in random local quantum circuits with charge conservation.

arXiv:1902.00977Feb 3, 201961 citations

Addendum to "Dynamics of Renyi entanglement entropy in local quantum circuits with charge conservation''

Yichen Huang

In this addendum to [Huang, arXiv:1902.00977], I explicitly present results for qudit systems, i.e., spin systems with constant local dimension $d\ge2$ .In this addendum to [Huang, arXiv:1902.00977], I explicitly present results for qudit systems, i.e., spin systems with constant local dimension $d\ge2$.

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Entanglement growth in diffusive systems with large spin

Tibor Rakovszky, Frank Pollmann, Curt W. von Keyserlingk

Recent works 1 , 2 argued that Rényi entropies S α with indices α > 1 exhibit a sub-ballistic, $\propto \sqrt{t}$ , growth in systems with diffusive transport. A subsequent work by Žnidarič 3 claims that such sub-ballistic growth occurs only in certain cases (in particular, d = 1 dimensional systems with q = 2 states per site) and is generically replaced by ballistic growth in the absence of further fine-tuning. Below, we try to make precise the conditions needed for diffusive (rather than ballistic) Rényi growth and argue that they apply to a much wider class of systems than what is suggested by Žnidarič 3 . In particular, the examples presented by Žnidarič that avoid diffusive growth do so because of the presence of additional non-conserved degrees of freedom, not merely their larger local Hilbert space.Recent works 1 , 2 argued that Rényi entropies S α with indices α > 1 exhibit a sub-ballistic, $\propto \sqrt{t}$ , growth in systems with diffusive transport. A subsequent work by Žnidarič 3 claims that such sub-ballistic growth occurs only in certain cases (in particular, d = 1 dimensional systems with q = 2 states per site) and is generically replaced by ballistic growth in the absence of further fine-tuning. Below, we try to make precise the conditions needed for diffusive (rather than ballistic) Rényi growth and argue that they apply to a much wider class of systems than what is suggested by Žnidarič 3 . In particular, the examples presented by Žnidarič that avoid diffusive growth do so because of the presence of additional non-conserved degrees of freedom, not merely their larger local Hilbert space.

Communications PhysicsMay 10, 20218 citations

In local quantum circuits with charge conservation, we initialize the system in random product states and study the dynamics of the Renyi entanglement entropy

R_α

. We rigorously prove that

R_α

with Renyi index

α>1

at time

t

\le O(\sqrt{t\ln t})

if the transport of charges is diffusive. Very recent numerical results of Rakovszky et al. show that this upper bound is saturated (up to the sub-logarithmic correction) in random local quantum circuits with charge conservation.

In this addendum to [Huang, arXiv:1902.00977], I explicitly present results for qudit systems, i.e., spin systems with constant local dimension

d\ge2

z

is identified as the dynamical exponent characterizing transport of the conserved charges. Here, however, we demonstrate that this simple identification may not hold in certain quantum systems with kinetic constraints. In particular, we study two types of U(1)-symmetric quantum automaton circuits with XNOR (exclusive NOR) and Fredkin constraints, respectively. We find numerically that while spin transport in both models is subdiffusive, the second R\'enyi entropy grows diffusively in the XNOR model, and superdiffusively in the Fredkin model. For systems with the XNOR constraint, this distinction arises since the spin correlation function can be attributed to an emergent tracer dynamics of tagged particles, whereas the R\'enyi entropies are constrained by collective transport of the particles. Our results suggest that care must be taken when relating transport and entanglement entropy dynamics in generic quantum systems with conservation laws.

Recent works 1 , 2 argued that Rényi entropies S α with indices α > 1 exhibit a sub-ballistic,

\propto \sqrt{t}

, growth in systems with diffusive transport. A subsequent work by Žnidarič 3 claims that such sub-ballistic growth occurs only in certain cases (in particular, d = 1 dimensional systems with q = 2 states per site) and is generically replaced by ballistic growth in the absence of further fine-tuning. Below, we try to make precise the conditions needed for diffusive (rather than ballistic) Rényi growth and argue that they apply to a much wider class of systems than what is suggested by Žnidarič 3 . In particular, the examples presented by Žnidarič that avoid diffusive growth do so because of the presence of additional non-conserved degrees of freedom, not merely their larger local Hilbert space.